An (n, d, k)-mapping f is a mapping from binary vectors of length n to permutations of length n + k such that for all x, y ∈ f>H (f(x), f(y))≥ d H (x, y)+ d, if d H (x, y)≥ (n + k)-d and d H (f(x), f(y)) n + k, if d H (x, y) > (n + k) - d. In this paper, we construct an (n,3,2)-mapping for any positive integer n ≥ 6. An (n, r)-permutation array is a permutation array of length n and any two permutations of which have Hamming distance at least r. Let P(n, r) denote the maximum size of an (n, r)-permutation array and A(n, r) denote the same setting for binary codes. Applying (n,3,2)-mappings to the design of permutation array, we can construct an efficient permutation array (easy to encode and decode) with better code rate than previous results [Chang (2005). IEEE Trans inf theory 51:359-365, Chang et al. (2003). IEEE Trans Inf Theory 49:1054-1059; Huang et al. (submitted)]. More precisely, we obtain that, for n≥ 8, P(n, r)≥ A(n-2, r-3)>A(n-1,r-2)= A(n, r-1) when n is even and P(n, r) ≥ A(n-2, r-2 3)=A(n-1, r-2)>A(n, r-1) when n is odd. This improves the best bound A(n-1,r-2) so far [Huang et al. (submitted)] for n ≥ 8.
- Coding theory
- Permutation array